This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Formula display:

Abstract

We show that certain three-dimensional (3D) superlattice nanostructure based on Bi2Te3 topological insulator thin films has better thermoelectric performance than two-dimensional
(2D) thin films. The 3D superlattice shows a predicted peak value of ZT of approximately 6 for gapped surface states at room temperature and retains a high
figure of merit ZT of approximately 2.5 for gapless surface states. In contrast, 2D thin films with
gapless surface states show no advantage over bulk Bi2Te3. The enhancement of the thermoelectric performance originates from a combination
of the reduction of lattice thermal conductivity by phonon-interface scattering, the
high mobility of the topologically protected surface states, the enhancement of Seebeck
coefficient, and the reduction of electron thermal conductivity by energy filtering.
Our study shows that the nanostructure design of topological insulators provides a
possible new way of ZT enhancement.

Background

is usually baffled by the competition of the Seebeck coefficient S, the electrical conductivity σ, the electron thermal conductivity κe and the lattice thermal conductivity κl. Recent discoveries that some of the best thermoelectric materials such as Bi2Te3[1] are also strong 3D topological insulators [3-5], and experimental studies of the mechanical exfoliation and growth of quintuple layers
(QL, 1 QL ≈ 0.748 nm) of Bi2Te3[6,7] attract much interest [8-11] in the thermoelectric properties of thin films of Bi2Te3 with one or a few QL.

High ZT values of Bi2Te3 thin films depend crucially on the opening of a subgap at the surface, which disappears
quickly with the increasing of the film thickness, as suggested both theoretically
[12-14] and experimentally [15,16]. Relatively accurate density functional theory calculations [16,17] show that the (indirect) surface gap of Bi2Te3 vanishes as soon as the thickness of the thin film increases to 3QL. Despite the
high mobility [18] of the surface electrons, the gapless surface states would lead to poor thermoelectric
performance due to low Seebeck coefficient and high electron thermal conductivity.
However, by creating suitable nanostructures, extra energy-dependent electron scattering
mechanisms can be introduced, which could increase the Seebeck coefficient [19,20] and reduce the electron thermal conductivity. The consideration of nanostructures
of thin films is also motivated by the fact that a single layer of thin film is not
of much practical use for thermoelectric applications, and stacks of thin films have
much lower lattice thermal conductivity compared with the bulk [21].

In this paper, we propose a 3D superlattice nanostructure based on thin films of Bi2Te3 topological insulator, with one thin film (thickness a) and one stripe-shaped layer (thickness c along the z direction and width b along the x direction) stacked alternatively; the spacing between two nearby stripes is d, as shown in Figure 1. The system is considered to be infinite along the y direction and periodic in the x and z directions. We are interested in thermoelectric transport along the x direction. Along the transport direction, the surface carriers will encounter potential
barriers/wells if the surface gap is different from the bulk gap. We will give a comparison
study of the thermoelectric properties of 2D thin films and the 3D superlattice and
show that the latter has higher ZT values due to the enhanced Seebeck coefficient and reduced electron thermal conductivity
through energy filtering as well as the reduced lattice thermal conductivity by interface
phonon scattering. More importantly, the 3D superlattice shows high ZT values even when the surface states are gapless, in which case 2D thin films have
low ZT values.

Figure 1.A sketch of the physical system to be studied. The system can be regarded as a 3D superlattice with one thin film (thickness a) and one stripe-shaped layer (thickness c along the z direction and width b along the x direction) stacked alternatively; the spacing between two adjacent stripes is d. The transport direction is along the x axis.

The 3D superlattice structure proposed here can be regarded as bulk Bi2Te3 which is nanoporous and resembles nanoporous Si [22] and nanoporous Ge [23], both of which show significant enhancement of the figure of merit due to orders
of magnitude reduction in the lattice thermal conductivity. The difference is that
the electronic transports for nanoporous Bi2Te3 and Si/Ge are dominated by surface and bulk carriers, respectively.

Methods

We first consider the lattice thermal conductivity which depends on the geometric
parameters and can be estimated using the modified effective medium approximation
[24] which treats the system as a nanocomposite. In our case, we can perceive the system
as a nanocomposite with vacuum (hole) included in the bulk phase. Taking the thermal
conductivity of the holes to be zero, we can express the effective lattice thermal
conductivity of the 3D superlattice along the x direction as

(2)

where Kb and Λb are lattice thermal conductivity and phonon mean free path (MFP) for the bulk phase,
respectively; is the MFP corresponding to the collision of the phonons onto the holes, and is the volume fraction of the holes. The derivation of Λb proceeds as follows: The effective area of collision for a phonon upon a rectangular
vacancy is cΔy, where Δy is an arbitrary length in the y direction; if a phonon travels at a distance L, it will encounter N = cΔyLn vacancies, where is the number density of the vacancies. The MFP Λc is thus . The well-defined values for the bulk phase phonon MFP can be extracted [25] from experimental values of the bulk lattice thermal conductivity [26,27] and phonon dispersions. Figure 2 exhibits significant reduction of the lattice thermal conductivity of the 3D superlattice
from that of bulk Bi2Te3. Note that the effective lattice thermal conductivity for the 3D superlattice is
weakly temperature-dependent, indicating the dominance of interface scattering. Our
result is qualitatively consistent with those from molecular dynamics simulations
[9] and experimental measurements [21] on similar nanostructures. Generally, Equation 2 overestimates the lattice thermal
conductivity since it ignores other possible phonon scattering channels. As for the
lattice thermal conductivity of a single thin film, we take it to be the same as the
bulk value, according to molecular dynamics simulation results [9].

Figure 2.Lattice thermal conductivities. Lattice thermal conductivities for bulk Bi2Te3 obtained by fitting of experimental data [26,27] and for the 3D superlattice with different geometric parameters calculated by Equation
2.

A comparison with nanoporous Si [22] and Ge [23] is helpful. While the values of lattice thermal conductivities for bulk Bi2Te3, Ge and Si range from a few watts per meter Kelvin to several hundred watts per meter
Kelvin, the values for the corresponding nanoporous materials are all reduced to below
1 W/(m K). This fact is another indication of the dominance of phonon-interface scattering
over the phonon-phonon scattering.

We then consider electron transport coefficients. Since the system is considered to
be infinite, the electronic transport is diffusive and Boltzmann's formalism applies.
By solving the Boltzmann transport equations with the relaxation time approximation,
one can express the thermoelectric transport coefficients as [28]

(3)

(4)

(5)

where e is the magnitude of the elementary charge, T is the absolute temperature, μ is the chemical potential, and the integrals (f is the Fermi-Dirac distribution)

which are determined by the electronic structure and the electron scattering mechanisms
of the material. For simple band structure, the TDF is a product of density of states
g(E), velocity square along the transport direction vx(E)2, and electron relaxation time τ(E):

(8)

The electronic structures of thin films of topological insulator differ significantly
from that of the bulk. First, as the thickness of the film is reduced to one or a
few QL, the spin-polarized surface states at one surface will mix up with the components
of the opposite spin from the other surface and lead to a hybridization gap at the
Dirac point to avoid the crossing of bands with the same quantum number [12-17]. Second, at this thickness, the layers underneath the film surface should be treated
as quantum well, which is indicated by experimental observations [15]. In this paper, we adopt a simple parameterization [8] of the dispersion relation of the surface states

(9)

where ‘±’denotes the conduction (valance) band, VD of approximately 4 × 105 m/s [4,18] is the Dirac velocity, and 2 ∆f is the surface gap. The zero of energy is chosen to be at the center of the surface
gap. This simple dispersion relation is derived from symmetry arguments [8] and exhibits the essentials of thermoelectric properties of the surface states. From
the above dispersion relation, we can express the velocity square along the transport
direction and the effective 3D density of states for a thin film with thickness a as

(10)

(11)

For the 3D superlattice structure, the square of velocity along the transport direction
takes the same form as in Equation 10, and the effective 3D density of states is that
of Equation 11 as scaled by a/(a + c). Since the quantum well states lie much above, we can safely disregard them and
consider the surface states only. We also only consider the conduction band with E > 0.

To compute the electronic transport coefficients, we should also find an estimation
of the electron relaxation time. Experimental studies [18] show that the surface electron mobility μs of Bi2Te3 approaches 104 cm2 V−1 s−1, about an order of magnitude higher than the bulk value. We use this experimental
value of mobility to calculate the intrinsic surface electron relaxation time τs. For the superlattice, the surface carriers also encounter potential barriers/wells
whenever they reach the boundaries of the surfaces (located at the crossing lines
between the thin films and the strips) and suffer from additional scattering. The
strips are modeled by rectangular potential barriers/wells with height Vi = Δb − Δf and width b and average distance L = b + d. Let the transmission probability for the charge carriers with energy E through a single strip be P(E). The path length after passing through the n-th strip and scattering by the (n + 1)-th one is nLP(E)n(1 − P(E)). The mean free path is the sum of all of the possible path lengths [31], ∑ n = 1∞nLP(E)n(1 − P(E)) = LP(E)/(1 − P(E)). The corresponding additional relaxation time τa is the mean free path divided
by the velocity,

(12)

The transmission probability P(E) is determined by the interface potential barrier/well according to the following
standard quantum mechanical calculations:

(13)

The total surface electron relaxation time τ for the superlattice is given by

(14)

This method of calculating the total electron relaxation time has been recently applied
to the study of thermoelectric properties of nanocomposites [32,33]. We assume 2Δb = 0.15 eV for bulk Bi2Te3 according to the experimental value [34]. The surface gap is chosen to be 2Δf = 0.3, 0.06, and 0 eV for thin films with thickness 1QL, 2QL, and 3QL, respectively,
as suggested by first-principle calculations [17]. The effective mass m* entering Equation 13 stands for that of bulk Bi2Te3, which has a highly anisotropic effective mass tensor, with the in-plane components
0.021 m0 and 0.081 m0 and the out-of-plane component 0.32 m0 (m0 is the mass of a free electron) [35]. For simplicity, we take m* to be 0.1 m0 in our calculations. The exact value of m* is not very crucial for our discussions, since it only affects the optimal value
of b.

Results and discussion

Figure 3 shows the calculated ZT values for 2D thin films and the 3D superlattice with varying geometric parameters.
Both of the stand-alone thin film with thickness of 1 QL and the 3D superlattice with
a = 1QL exhibit high figure of merit ZT of 5 to 6 at room temperature with appropriate doping level. For thicker films, the
surface gap becomes smaller and yields lower ZT values. However, the 3D superlattice is much more robust against the disappearance
of surface gap and still shows peak value of ZT of approximately 2.5 (Figure 3f) for gapless states. In contrast, a gapless thin film with a thickness of 3QL demonstrates
no better performance than the bulk material (Figure 3e).

Figure 3.Figure of merit. ZT values as functions of temperature and chemical potential for Bi2Te3 2D thin films with thickness 1QL (a), 2QL (c), and 3QL (e); and the 3D superlattice with a = 1QL (b), 2QL (d) and 3QL (f). For the a = 1QL case, c = 1QL, b and d = 10 nm; for the a = 2QL case, c = 2QL, b = 7.5 nm and d = 10 nm; for the a = 3QL case, c = 3QL, b = 7.5 nm and d = 10 nm. The zero of energy is chosen to be at the center of the surface gap.

ZT values depend on the geometric parameters a, b, c and d since both the lattice thermal conductivity and the electronic transport coefficients
depend on these parameters. The dependence of ZT on the geometric parameters for a = 3QL case is shown in Figure 4. While ZT has a weak dependence on d, it depends strongly on b since this parameter largely affects the transmission probability P(E) and hence the electron relaxation time. Only an appropriate b gives the desired energy filtering effect. The weak dependence of ZT on d results from the fact that the phonon collision probability 1/Λc and the hole volume fraction ϕ have opposite dependences on d and the fact that the total electron relaxation time is not strongly dependent on
d. The optimized value of c results from the optimization of the B factor [36] which represents the relative transport strength of electrons over phonons. A large
value of c gives a lower lattice thermal conductivity and a lower effective 3D density of states,
and only an appropriate value of c gives an optimized B factor. We should note that our model is only valid for appropriate ranges of these
geometric parameters. For example, our model cannot be extrapolated to the c = 0 case, since in this limiting case, our model treats the system as parallel thin
films with vanishing separation rather than the bulk material. The conclusion of this
parametric study leads to the following recommendations for the relevant parameters:
c takes the same value of a, b takes the value of 5–10 nm, and d takes the value of 10–20 nm.

Figure 4.Optimized geometric parameters. Optimization of the geometric parameters b, c, and d for the 3D superlattice with a = 3QL at T = 250 K and μ = 60 meV. The geometric parameter dependence of ZT values for 3D superlattice with a = 2QL and 1QL has similar trends as given here for the case of a = 3QL.

To understand why 3D superlattice outperforms 2D thin films, we plot in Figure 5 the individual thermoelectric transport coefficients as functions of chemical potential
at T = 250 K for the case of a = 3QL, where the difference of the ZT values is most significant. As expected, the electrical conductivity of the superlattice
is heavily reduced (Figure 5a) due to the additional scattering by the potential barriers/wells. The additional
scattering reduces the effective mobility of charge carriers and decreases the electrical
conductivity. This reduction of electrical conductivity is not desirable for obtaining
high figure of merit. However, this additional scattering also has two beneficial
effects on thermoelectric transport, which result in an increase of the Seebeck coefficient
and a decrease of the electronic thermal conductivity. Firstly, the additional relaxation
time is strongly energy dependent in such a way that charger carriers with energies
lower than a certain value are largely scattered back and those with energies higher
than a certain value mostly transport through. With an appropriate choice of the barrier/well
width b and the chemical potential, one can effectively filter out the charge carriers with
energies lower than the chemical potential. Since charger carriers with energies above
and below the chemical potential contribute oppositely to the Seebeck coefficient
(Equation 4), this energy-filtering mechanism can significantly enhance the Seebeck
coefficient (Figure 5b). Whether the power factor is increased or decreased due to this energy filtering
effect depends on the choices of relevant parameters. In the present case, the power
factor is decreased (Figure 5c). From Figure 3e,f, we can infer the optimal chemical potential as about 65 meV. At this value of chemical
potential, the power factor for the 2D thin film is about twice as large as that for
the 3D superlattice. Then, why does the figure of merit for superlattice reach a value
of about 2.5, while that for the thin film only takes a value of about 0.6? This is
mainly resulted from the significant reduction of the electronic thermal conductivity
for the superlattice compared with the thin film. As can be seen from Figure 5d, the electronic thermal conductivities for the thin film and the superlattice are
about 13 and 1 W/(m K) respectively. Combined with the reduction of the lattice thermal
conductivity, the total thermal conductivity for the superlattice is about 1/8 of
that for the thin film. A combination of all these effects results in a four-fold
enhancement of the ZT for the superlattice compared with the thin film. The significant reduction of the
electronic thermal conductivity also results from the energy filtering mechanism.
With the filtering of the low-energy charger carriers, the energy distribution of
transport distribution function [29,30] becomes more concentrated, resulting in a violation of the Wiedemann-Franz law [28] which states that the electronic thermal conductivity is proportional to the electrical
conductivity, κe = L0σT, with the Lorentz number WΩK−2. From another point of view, this violation makes the electronic transport more reversible
[37], which is desirable for efficient thermoelectric energy conversion.

To get an intuition of the necessity of violating the Wiedemann-Franz law for superior
thermoelectric performance, suppose that the lattice thermal conductivity is reduced
to zero, and the Seebeck coefficient is 200 μVK−1, then if the Wiedemann-Franz law were strictly valid, we would have achieved a relatively
low figure of merit ZT = S2/L0 ≈ 1.6 regardless how large the electrical conductivity would be. Thus, as we approach
the lower limit of lattice thermal conductivity, it is imperative to find a way to
change the shape of the transport distribution function [29,30] either by altering the electronic structures [38] or by introducing energy-dependent electron scattering mechanisms.

Finally, we add some view points on the approach that we used in this work. For the
study of thermoelectric transport of a nanostructured material, there are two complementary
ways of viewing the system. One is to take the system as a whole, in which case the
nanostructures do alter the electronic structure of the system, but it is difficult
to calculate the band structure of such large system directly by first principles
method due to large number of atoms presented in nanostructures. Another way is to
view the system as some bulk material with nanostructures that do not affect the electronic
structure of the bulk material significantly, but introduce some extra scatterings
for the charge carriers. We have chosen the second approach in our study. This approach
has been widely used in the community of thermoelectrics. For example, in the study
of nanocomposites with grain boundaries [32,33], one usually assumes that the electronic structure inside the grain boundary is the
same as that of the corresponding bulk material. The grain boundary does not affect
significantly the energy-band structure and only serves as a scattering interface.
The only difference between our model and the nanocomposite models [32,33] is that our bulk material is quasi-two-dimensional instead of three-dimensional,
and the grain boundaries are replaced by the strips in our proposed structure. So
long as the average distance between the strips is large compared with the size of
the strips (which is the case for the optimized structures), this view point is valid
and there is no significant deficiency in our model.

Conclusions

In summary, we demonstrated that certain nanostructures of topological insulators
have the potential of overcoming the obstacle of competition of the individual thermoelectric
transport coefficients to achieve high thermoelectric figure of merit. High electron
mobility of the topologically protected surface states together with the holy structure
of the 3D superlattice ensures a large B factor [36], and the energy filtering effect introduced by the inhomogeneous superlattice structure
promotes the Seebeck coefficient and the ratio of electrical conductivity to the electron
thermal conductivity. The optimal temperature of performance for the 3D superlattice
with optimized geometric parameters is around or below room temperature, making it
very appealing for thermoelectric power generation and refrigeration applications
around and below room temperature, respectively. In addition, a similar structure
has appeared in a thin film transistor array, with an insulating substrate and a stripe-shaped
semiconductor layer for a plurality of transistors [39], which demonstrates the experimental feasibility of our proposed 3D superlattice
structure. The detailed information of geometric and electronic properties of the
fabricated superlattice can be characterized by integrated electron scattering and
X-ray scattering techniques [40,41].

Competing interests

The authors declare that they have no competing interests.

Authors' contributions

ZF carried out the main part of the calculations and drafted the manuscript. JZ carried
out part of the calculations. H-QW participated in the organization of the project
and discussions of the results and revised the manuscript. J-CZ organized the project,
analyzed the results and revised the manuscript. All authors read and approved the
final manuscript.

Acknowledgements

This work is supported by the Minjiang Scholar Distinguished Professorship Program
through Xiamen University of China, Specialized Research Fund for the Doctoral Program
of Higher Education (grant numbers 20090121120028 and 20100121120026), Program for
New Century Excellent Talents in University (NCET) (grant number NCET-09-0680) and
the National Science Foundation of China (grant number U1232110).